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I have a dataset for gym memberships over 5 years. I'm trying to predict what is causing people to cancel their policies. Variables I have include age, gender, income, locations... all demographical. These variables stay the same between each year, but I also have a variable called membership fee change... which is a percentage change in their membership fee, which changes each year.

If a member renews all 5 years, they'll appear in the dataset 5 times with the dummy indicator taking a value of 1. If they cancel in year 2, they'll appear twice... year 1's record will have a 1 for the dummy variable but year 2's will be 0.

What kind of regression would be most suitable to predict cancellations?

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Whether you should use Cox regression, logistic regression or some other method depends on exactly what you are trying to do.

You write

I'm trying to predict what is causing people to cancel their policies

and also

What kind of regression would be most suitable to predict cancellations

So it's not clear whether you are primarily interested in explanation (first statement) or prediction (second statement).

In addition, you need to decide whether you are interested in a) whether the person cancels at all (which would indicate that some form of logistic regression should be used, possibly a non-linear multilevel model to deal with the changing fees, or you might have some other way to deal with those changes, or, if the changes happened to all members, you might need to ignore them) or b) how long the person stays a member, which would indicate some form of survival analysis, possibly Cox regression but perhaps some other form.

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